In logic, the scope of a quantifier or connective is the shortest formula in which it occurs, [1] determining the range in the formula to which the quantifier or connective is applied. [2] [3] [4] The notions of a free variable and bound variable are defined in terms of whether that formula is within the scope of a quantifier, [2] [5] and the notions of a dominant connective and subordinate connective are defined in terms of whether a connective includes another within its scope. [6] [7]
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The scope of a logical connective occurring within a formula is the smallest well-formed formula that contains the connective in question. [2] [6] [8] The connective with the largest scope in a formula is called its dominant connective, [9] [10] main connective, [6] [8] [7] main operator, [2] major connective, [4] or principal connective; [4] a connective within the scope of another connective is said to be subordinate to it. [6]
For instance, in the formula , the dominant connective is ↔, and all other connectives are subordinate to it; the → is subordinate to the ∨, but not to the ∧; the first ¬ is also subordinate to the ∨, but not to the →; the second ¬ is subordinate to the ∧, but not to the ∨ or the →; and the third ¬ is subordinate to the second ¬, as well as to the ∧, but not to the ∨ or the →. [6] If an order of precedence is adopted for the connectives, viz., with ¬ applying first, then ∧ and ∨, then →, and finally ↔, this formula may be written in the less parenthesized form , which some may find easier to read. [6]
The scope of a quantifier is the part of a logical expression over which the quantifier exerts control. [3] It is the shortest full sentence [5] written right after the quantifier, [3] [5] often in parentheses; [3] some authors [11] describe this as including the variable written right after the universal or existential quantifier. In the formula ∀xP, for example, P [5] (or xP) [11] is the scope of the quantifier ∀x [5] (or ∀). [11]
This gives rise to the following definitions: [a]
In logic, the scope of a quantifier or connective is the shortest formula in which it occurs, [1] determining the range in the formula to which the quantifier or connective is applied. [2] [3] [4] The notions of a free variable and bound variable are defined in terms of whether that formula is within the scope of a quantifier, [2] [5] and the notions of a dominant connective and subordinate connective are defined in terms of whether a connective includes another within its scope. [6] [7]
Logical connectives | ||||||||||||||||||||||
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Related concepts | ||||||||||||||||||||||
Applications | ||||||||||||||||||||||
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The scope of a logical connective occurring within a formula is the smallest well-formed formula that contains the connective in question. [2] [6] [8] The connective with the largest scope in a formula is called its dominant connective, [9] [10] main connective, [6] [8] [7] main operator, [2] major connective, [4] or principal connective; [4] a connective within the scope of another connective is said to be subordinate to it. [6]
For instance, in the formula , the dominant connective is ↔, and all other connectives are subordinate to it; the → is subordinate to the ∨, but not to the ∧; the first ¬ is also subordinate to the ∨, but not to the →; the second ¬ is subordinate to the ∧, but not to the ∨ or the →; and the third ¬ is subordinate to the second ¬, as well as to the ∧, but not to the ∨ or the →. [6] If an order of precedence is adopted for the connectives, viz., with ¬ applying first, then ∧ and ∨, then →, and finally ↔, this formula may be written in the less parenthesized form , which some may find easier to read. [6]
The scope of a quantifier is the part of a logical expression over which the quantifier exerts control. [3] It is the shortest full sentence [5] written right after the quantifier, [3] [5] often in parentheses; [3] some authors [11] describe this as including the variable written right after the universal or existential quantifier. In the formula ∀xP, for example, P [5] (or xP) [11] is the scope of the quantifier ∀x [5] (or ∀). [11]
This gives rise to the following definitions: [a]